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Questions tagged [differential-geometry]

Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses on "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such structures, use (differential-topology) instead. Use (symplectic-geometry), (riemannian-geometry), (complex-geometry), or (lie-groups) when more appropriate.

Differential geometry is a branch of mathematics that uses techniques in calculus and linear algebra to study geometry. It is closely related to differential topology and PDEs (geometric analysis).

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Is the hemisphere bendable?

There is a theorem in classical differential geometry due to Cohn Vossen (1927) that compact, connected regular surfaces of everywhere positive gaussian curvature (ovaloids) are rigid - in the sense that if a regular surface in Euclidean space is…
Artin Ostal
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Simplification in the curvature tensor identity

In Milnor's Morse Theory, he defines the curvature $R$ of an affine connection $\vdash$ as $$R(X,Y)Z=-X\vdash(Y\vdash Z)+Y\vdash(X\vdash Z)+[X,Y]\vdash Z.$$ In his proof of Lemma 9.3 part (2), which states that $R(X,Y)Z+R(Y,Z)X+R(Z,X)Y=0$, he writes…
boink
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Diffeomorphism preserves dimension

I read from Milnor's book $\textit{Topology from the Differentiable Viewpoint}$ this assertion "If $f$ is a diffeomorphism between opensets $U\subset R^k$ and $V\subset R^l$, then k must equal l, and the linear mapping $$df_x:R^k\rightarrow R^l$$…
Evariste
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Show a certain vector field $V$ only commutes with others that are collinear with $V$

I found this on an old qualifying exam. I started the problem, but I'm not sure what my next step should be: Let $T^2$ be the standard 2-dimensional torus with $\mathbb{Z}$-periodic coordinates $(x,y)$. Consider the vector field $$ V = \sin(2\pi y)…
phenomenalwoman4
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Homeomorphism between convex set and a sphere

For $C$ a compact convex set of $\mathbb{R}^n$ $(n\geq 1)$ whose interior is nonempty and whose boundary $\partial C$ is a smooth manifold, is $\partial C$ homeomorphic to the sphere $\mathbb{S}^{n-1}$?
Gregoire Panel
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$X(u, v) = (u + v, u + v, uv)$, where $U = \{(u, v)\in R^2\; |\; u> v\}$ is parametrisation for $P = \{(x, y, z) \in R^3\;|\; x = y\}$?

Let $P = \{(x, y, z) \in R^3\;|\; x = y\}$ (a plane) and let $X: U \subset R^2\to R^3$ be given by $X(u, v) = (u + v, u + v, uv)$, where $U = \{(u, v)\in R^2\; |\; u> v\}$. P. Is $X$ a parametrisation of $P$? I know that $P$ is a plane and so it is…
chesslad
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Integral over surface vector in the language of differential geometry?

I would like to relate the idea of taking an integral over a surface vector in the language of differential geometry. For example, the integral (in $\mathbb{R}^3$) $$ \vec{A} = \int_{d\Omega} f d\vec{S} $$ where $f: \mathbb{R}^3 \rightarrow…
user1576720
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For two parameterizations, normal unit vector coincides if the derivative of transition map has positive determinant

Let $S\subset R^3$ be a regular surface with $p\in S$. Let $(U,\phi)$ be a local chart such that $\phi(q) = p$ where $q\in U\subset R^2$ (Basically, $(U,\phi)$ is a local chart when $\phi$ is a local parameterization of $S$ that has $p$ in its image…
chesslad
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Diffeomorphism between ellipsoid and sphere?

I know that a diffeomorphsim between the sphere $S^2 = \{(x, y, z) \; | \; x^2+y^2 +z^2= 1\}$ and the ellipse $E^2 = \{(x, y, z) \; | \; (\frac{x}{a})^2+(\frac{y}{a})^2 +(\frac{z}{a})^2= 1\}$ is given by $$f:S^2\to E^2 \;\;\ni\;\; (x,y,z) \mapsto…
chesslad
  • 2,533
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Pullback metric of inversion map

Let $I:\mathbb{R}^{n}\setminus\{0\} \longrightarrow \mathbb{R}^{n}\setminus\{0\}$ be the inversion map, i.e. $I(x)=\frac{x}{|x|^{2}}$ and equip $\mathbb{R}^{n}\setminus\{0\}$ with the standard euclidean metric $g_{eucl}=\langle\cdot,\cdot\rangle$.…
Giuliano Cantina
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How does the signed curvature $k_s$ changes by applying the dilation $v\mapsto av, \;\; a\neq 0$

I am having trouble solving Exercise 2.2.4 of Andrew Pressley's Elementary Differential Geometry [2.2.4] Let $k$ be the signed curvature of a plane curve $C$ expressed in terms of its arc-length. Show that, if $C_a$ is the image of C under the…
chesslad
  • 2,533
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Prove $M(\alpha(-s)) = \alpha(s)$ for all s in $(-\epsilon,\epsilon)$, where M: $\mathbb{R}^2 \rightarrow \mathbb{R}^2$

Let $\epsilon>0$ and $\alpha:(-\epsilon,\epsilon) \rightarrow \mathbb{R}^2$ be a regular plane curve parametrized by arc-length. Suppose that $k(s) = k(-s)$ for all $s \in (-\epsilon,\epsilon)$. Prove that $M(\alpha(-s)) = \alpha(s)$ for all $s$ in…
Ju Grigori
  • 385
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Volume in the unit sphere bundle

Consider $M$ a riemannian manifold and $A \subset M$. Let $SM$ the sphere bundle of $M$. There exist any relation among the volume of $A$ in $M$ and $\pi^{-1}(A)$ in $SM$?. I guess that as the fibers are…
JulianDoyle
  • 387
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Is this a sufficient condition for a curve to be perpendicular to a fixed point?

Suppose we have a fixed point $p \in \mathbb{R}^3$ and a smooth curve $\beta \colon I \to \mathbb{R}^3$ such that $\beta(0)\perp p$ and $\beta'(t) \perp p $ for all $ t \in I$. Is this condition enough to ensure that $\beta(t) \perp p$ for all $t…
Elma
  • 320
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Can $\frac{\tau}{\kappa}$ be simplified in Differential Geometry?

I'm working through some problems in Differential Geometry but I always get stuk at this expression. One of the problems for instance asks to calculate the arclength paramater of the following function: $\epsilon: J\rightarrow \mathbb{R}^3: s…
Othman El Hammouchi
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